In particular, frames are widely used in sampling theory in [1]} amounts to the construction of Banach frames consisting of reproducing kernals for a large class of shift invariant spaces. Aldroubi et al. [2]} used Banach frames in various irregular sampling problems. Elder and Forney [3]} used tight frames for quantum measurement. Gröchenig [4]} emphasised that localization of a frame is a necessary condition for its extension to a Banach frame for the associated Banach spaces. He also observed that localized frames are universal Banach frames for the associated family of Banach spaces.
Fornasier [5]} studied Banach frames for \(\alpha \) -modulation spaces. In fact, he gave a Banach frame characterization for the \(\alpha \) -modulation spaces. Shah et.al [6]} defined and studied Banach frames to a new geometric notation; in fact they gave a sufficient condition and a necessary condition for a cone associated with a Banach frame to be a generating cone.
